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T H E C O N S T... T O P A R T I A...
```Volume 114A, number 3
PHYSICS LETTERS
10 February 1986
T H E C O N S T R U C T I O N OF S P E C I A L S O L U T I O N S
TO PARTIAL DIFFERENTIAL EQUATIONS
Peter J. O L V E R 1
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
and
Philip R O S E N A U
Department of Mechanical Engineering, Technion, Haifa 32000, Israel
Received 9 November 1985; accepted for publication 26 November 1985
Almost all the methods devised to date for constructing particular solutions to a partial differential equation can be viewed
as manifestationsof a single unifying method characterized by the appending of suitable "side conditions" to the equation, and
solving the resulting overdetermined system of partial differential equations. These side conditions can also be regarded as
specifying the invariance of the particular solutions under some generalized group of transformations.
In the study of partial differential equations, the
discovery of explicit solutions has great theoretical
and practical importance. In the case of linear systems,
general solutions can be built up by superposition
from separable solutions; for nonlinear systems, explier solutions are used as models for physical or
numerical experiments, and often reflect the asymptotic behavior of more complicated solutions. Over
the years, a variety of methods for finding these special
solutions by reducing the partial differential equation
to one or more ordinary differential equations have
been devised. Included are the method of group-invariant solutions popularized by Ovsiannikov [1 ], but
due originally to Lie [2] (see also ref. [11] ),the "nonclassical method" for group-invariant solutions due to
Bluman and Cole [3], and its recent generalization by
the authors [4], the method of partially invariant solutions of Ovsiannikov [1 ], and the general method of separation of variables, for both linear systems as well as certain nonlinear equations such as Hamilton-Jacobi equations. The one common theme of all these methods
has been the appearance of some form of group invariance.
The purpose of this note is to explain how all the
above methods, as well as many others, can all be unified and significantly generalized by the concept of a
differential equation with side conditions. By this we
mean that to determine special solutions to a given
system of partial differential equations one proceeds
by appending one or more auxiliary differential equations, which we call "side conditions". The solutions
themselves will then be found by solving the entire
system of differential equations consisting of both
the original system along with the prescribed side conditions. The main difference between the various special methods devised for Finding explicit solutions is
then only the form or complexity of the relevant side
conditions. The most important conclusion to be
drawn from this approach is that the unifying theme
behind f'mding special solutions to partial differential
equations is not, as is commonly supposed, group
theory, but rather the more analytic subject of overdetermined systems of partial differential equations.
Thus the key question becomes not which groups are
relevant t o a given system o f partial differential equations, but rather which side conditions are admissible,
thereby providing genuine solutions of the system ?
1 Supported in part by NSF Grant MCS 81-00786.
What is now required is an algorithmic method of de107
Volume 114A, number 3
PHYSICS LETTERS
termining these compatible side conditions, and then
the corresponding special solutions. In this light, the
group-theoretic methods alluded to above can be regarded as special techniques that allow one to construct
particular classes of compatible side conditions; the
general set of side conditions is much larger. This is
not to say that the proposed method is meant to supplant the popular group-theoretic methods in use, but
rather to be utilized as a unifying frame-work in which
to compare and interpret all these different techniques.
Nor does it preclude the discovery of other formalisms
for constructing special solutions, although it certainly can provide motivation for the development of additional new classes of methods for determining explicit solutions. It also helps explain recent result of
Kalnins and Miller [5,6] on separation of variables,
ha which the group-theoretic interpretation relevant
to simpler systems seems to be no longer valid.
Actually, one can provide a group-theoretic explanation of the side conditions provided one uses the
theory of generalized symmetries (also known as LieBa'cklund transformations), although in many cases
this appears to us as a somewhat artificial re-interpretation of the basic issue. Using this point of view,
the side conditions relevant to group-invariant solutions come from ordinary geometrical groups of transformations, those for partially invariant solutions
from first order generalized symmetries which are not
equivalent to geometrical symmetries and those for
separable solutions from second order generalized
symmetries. Higher order symmetries lead to yet more
general types of ansatz for special solutions. What h a s
been lost is any underlying symmetry connection
with the system of partial differential equations itself
- there are no a priori restrictions on the groups under
symmetry case.)
Rather than try to develop a general theory here,
we have chosen to illustrate the basic concepts by a
series of examples of the different methods for constructing solutions, each reinterpreted in the light of
the unifying concept of a differential equation with
side conditions. Using these as a launchpad, the astute
reader will no doubt be able to envisage the form
which the general theory must take. Besides, in the
words of de Tocqueville, "God doesn't need general
theories - He knows all the special cases!"
We begin with Lie's classical theory of group-in108
10 February 1986
variant solutions, illustrated by the similarity solution
of the heat equation
(1)
Ut = U x x .
Consider the one-parameter group of scaling transformations (x, t, u) ~ (kx, h2t, u), h > 0. This is a classical symmetry group of the heat equation, in the
sense that it takes solutions to solutions. The condition that a solution be invariant under this group (i.e.
a "similarity solution") can be expressed in differential
form
xu x + 2tu t
= 0.
(2)
Thus the similarity solutions to the heat equation can
be determined as the solutions to the overdetermined
system consisting of the heat equation itself (1) together with the side condition (2) reflecting the sealhag invariance of the desired solutions. To solve this
system (1), (2), one can proceed to first solve (2)
using the method of characteristics, leading to the
fact that u = u ( x / x f f ) is a function of the similarity
variable ~ = x/x/Tonly. Substituting into (1), we see
that u satisfies the ordinary differential equation
u" + ~u'/2
= 0,
primes denoting derivatives with respect to ~. This
leads to the general scale-invariant solutions
u = c 1 erf(x/2x/rt) + c 2 ,
where erf is the standard error function.
In the nonclassical method introduced by Bluman
and Cole, one does not require that the group be a
symmetry group of the original system, but, less restrictively, that it be a symmetry group of the system
supplemented by the side conditions prescribing the
group invariance of the desired solutions. Rather than
treat the heat equation, since all the solutions obtained by the non-classical method already appear among
the classical group-invariant solutions in this case, cf.
ref. [3], we will use a less trivial example. Consider
the nonlinear wave equation
u t t = UUxx ,
(3)
whose classical symmetry group has been computed
ha ref. [7], p. 301. The one-parameter group G with
inf'mitesimal generator o = 2 t b x + 0 t + 8tO u does not
appear among the classical symmetries, and so is not
a candidate for the usual method of finding group-
Volume 114A, number 3
PHYSICS LETTERS
invariant solutions. Nevertheless, we can rind G-invariant solutions as follows. A function u = f(x, t) is
invariant under the group generated by u if and only
if it satisfies the side condition
(4)
8t = 2tu x + u t .
One easily checks that the combined pair of differential equations (3), (4) is invariant under the group G.
This is precisely the requirement needed to apply the
non-classical method, and hence we can Fred G-invafiant solutions to (3) by solving an ordinary differential
equation. The general solution of (4) is
u = 4t 2 + w ( x
-
t2),
where w depends on the invariant ~ = x - t 2. Substituting into (3) we see that w must satisfy the ordinary
differential equation
ww" + 2w' = 8,
where primes indicate derivatives with respect to ~.
This last equation can be integrated by Lie's method
for ordinary differential equations using the obvious
scaling and translational symmetries, but we will not
pursue this here. For each solution w = h(~), we obtain an explicit G-invariant solution u of the nonlinear wave equation (3); most of these do not appear
among the group-invariant solutions computed using
the ordinary symmetry groups of (3), and are thus
genuinely new invariant solutions not obtainable by
the classical method.
In ref. [4] it is shown how to generalize this
method to an arbitrary group of transformations on
the underlying space of independent and dependent
variables. For example, returning to the heat equa.
tion (1), the one-parameter group G generated by
the vector field o = tO t - x O x - 3 x 3 0 u is not an ordinary symmetry group of the heat equation. Nor is
it of the form amenable to the non-classical method
given by Bluman and Cole [3]. (Indeed, using their
notation on p. 1041, we would have X = - x / t , U =
- 3 x 3 / t [eL their eq. (90)], but these two functions
do not satisfy their defining equations (94)-(96).)
Nevertheless, there do exist G-invariant solutions of
the heat equation, and we can construct them as fol.
lows. The relevant side condition is
tu t - xu x
+3x 3 =0,
whose general solution is
10 February 1986
u = x3 + w ( x t ) ,
where w is a function of ~ = x t . Substituting into the
heat equation, we obtain
xw'
= t 2 w ''
+ 6x,
(5)
which is n o t an ordinary differential equation for w
as a function of ~. (Indeed, if it were, then G would
necessarily satisfy Bluman and Cole's non-classical
conditions!) However, treating x and ~ as independent
variables (valid provided t #: 0), we have
x3(w ' -
6) = ~2w" .
Since w is a function of ~ only, this latter equation is
satisfied for all x and ~ if and only if w satisfies the
pair of ordinary differential equations
w" = 0,
w' = 6 .
These are compatible, with solution w = 6~ + c, where
c is an arbitrary constant. Thus we obtain a one-parameter family of G-invariant solutions to the heat equation
u=x3+fxt+c,
which do not appear among the classical group-invariant solutions (although they are, of course, linear
combinations of two such solutions).
Before leaving side conditions arising from groupinvariance of solutions, it is worth pointing out that if
one has found a solution to a system of differential
equations, one can always devise a group that will lead
to the given solution by an application of the above
method. This is because any function u = f(x, t) is invariant under a multitude of groups acting on the
space of variables (x, t, u), and, as the generalized nonclassical method of ref. [4] does not impose a n y conditions on the group, any one o f these groups will lead
to the given solution. However, this reasoning is more
than likely done a p o s t e r i o r i , and therefore of limited
practical importance. Of more interest is the question
of which groups G lead to actual solutions; in other
words, when are the side conditions expresing the Ginvariance of a solution compatible with the system
under consideration. Obviously not any group will do,
but the problem of determining precisely which ones
are valid is no doubt very difficult, even for the simplest systems. (Note that it is even possible to devise
simple examples of classical symmetry groups of a sys109
Volume l14A, number 3
PHYSICS LETTERS
tern, to which no G-invariant solution can be found.
For instance the equation u t + u x = 1 admits the oneparameter group (x, t, u) ~ (x + e, t + e, u), e E R,
but there are no solutions to it which are invariant
under this group.)
Partially invariant solutions can also be treated by
this general approach, using side conditions in the form
of first order differential equations, but which are not
of the form prescribing group-invariance in the classical sense. (Indeed the general method of Ovsiannikov
rests on the theory of over-determined systems of differential equations!) As an illustrative example we
consider the system
Uy = Ox ,
uu x = Oy,
(6)
describing the transonic flow of a gas, which is treated
by Ovsiannikov (ref. [1], p. 286); note that these are
equivalent to the nonlinear wave equation uyy =
½(u2)xx (see below). For the two-parameter symme~y
group of translations (x, y, u, v) ~- (x + e, y, u, v + e)
generated by bx and 3v, a partially invariant solution
of rank 1 and defect 1 has general form u = q~(y), o =
~ ( x , y ) . Equivalently, we can prescibe this class of solutions by appending the single side condition
ux = 0 .
(7)
The resulting over-determined system (6), (7) has the
general solution
10 February 1986
metry group method. The partially invariant solutions
considered by Ovsiannikov (ref. [4], p. 286), have a
similar interpretation using the side condition x u x +
yUy = 0 reflecting their invariance under the generalized vector field (xu x + y U y ) a u . It is also a relatively
easy matter to extend Ovsiannikov's method to include solutions which are partiaUy invariant under
non-classical or even more general transformation
groups.
The third major class of special solutions to partial
differential equations are those obtained through separation of variables. The simplest form of separation
of variables is additive separation, and indeed all other
modes of separation known to date can, by a suitable
transformation, be reduced to additive separation.
For example, in the case of the heat equation (1), we
look for solutions of the form
u(x, t) = f ( x ) + g ( t ) .
(8)
Substituting (8) into (1), we are left with two ordinary
differential equations
f ' = 2~,
g'=•,
in terms of a separation constant X. This immediately
leads to the three-parameter family of additively separable solutions of the heat equation:
u ( x , t) = c2x2 + e l x + 2 c 2 t + e 0 ,
which is the most general such partially invariant solution. Note that we could reinterpret (7) as expressing
the invariance of the desired solutions under the oneparameter generalized group generated by the generalized vector field o = Uxa u ; the corresponding group
transformations are obtained by solving the system of
evolution equations
where ~, = 2c 2. As with partially invariant solutions,
the ansatz (8) does not arise from the invariance of
the separable solutions under some geometrical group
of transformations on the space of variables (x, t, u).
(Indeed, this would lead to only a single ordinary differential equation for the invariant solutions, and here
we have two ordinary differential equations.) However,
we can recover (8) from the second-order side condition
au/ae = u x ,
Uxt = 0,
U=clY+C 0 ,
O=ClX+C 2 ,
avlae = O ,
with corresponding group law
(u(x,y), o(x, y)) ~ (u(x + e, y), ~(x, y ) ) .
(See refs. [2,8] for the general theory of generalized
symmetries.) Note that this group is not equivalent to
a geometrical group acting locally on the variables (x,
y, u, o), but is truly "non-local". Also, the group is
not a symmetry group of the system (6) per se, so we
are in a "generalized version" of the non-classical sym110
(9)
in other words, additively separable solutions of the
heat equation are found by solving the system consisting of the heat equation (1) along with the second
order side condition (9). As with (7), we can interpret
(9) as requiring the desired solutions to be invariant
under the generalized symmetry group with infinitesimal generator n = u x t a u; the corresponding non-local
group transformations are found by solving the associated evolution equation
Volume 114A, number 3
Ou/ae = Uxt,
PHYSICS LETTERS
ul,~o = Uo(X,
t).
(10)
If the initial data u 0 is separable, then the solution to
(10) does not depend on the group parameter e, reflecting its invariance under the group. The recent
work of Kalnins and Miller [5,6] on additive separation for linear equations is, essentially, a detailed analysis of these special types of side conditions.
Multiplicative separation for the heat equation
comes from the ansatz
u(x, t) = f(x)g(t).
(11)
This can be reduced to additive separation by rewriting
the heat equation in terms of o = log u. Alternatively,
one can characterize (11) via the side condition
(12)
UUxt = UxU t .
(See ref. [9] for a more precise statement.) Combining
(1) and (12), we immediately deduce that a common
solution u(x, t) must satisfy the pair of ordinary differential equations
llUxx X = llXllXX ,
Ugltt = 122 ,
the latter following from differentiating (12) with respect to t. These can both be integrated once, leading
to the more familiar separation equations
Uxx+~U=O,
where (1) has required both integration constants to
be the same separation constant X. For X > 0 we recover the standard solutions
u = e -xt sin(x/~x),
to the heat equation. Again, we could reinterpret (12)
as specifying the invariance of these solutions under
the generalized symmetry group generated by ~ =
(UUxt - UxUt) Ou •
A more interesting, "nonclassical" form of multi.
plicative separation can be found in the equation
utt = Uxx + ~(U2)xx + "rUxxtt,
t3(w2)xx = X(w - VWxx),
both of whose solutions can be explicitly written in
terms of elliptic functions. (To integrate the second
equation, set ~ = w + ?~7/2~, and multiply by (~2)x x =
2w"wx.) Included among these are the elementary
rational solutions
u = ( I / 2 ~ ) ( [ ( x + c) 2 + 3q,]/(t + d ) 2 - l } ,
where c and d are arbitrary constants. An interesting
question is whether these simple solutions, which
appear among the solutions invariant under the nonlocal group generated by the vector field u = (UUxt UxU t + Uxt/2~) au, could, in fact, have been found
by local (but possibly non-classical) group methods.
Of course, according to ref. [4], any one of these
solutions could be found by use of some local group,
but it is not too hard to see that the entire two-parameter family could not have come from a single
local group. Indeed, suppose they were simultaneously invariant under some one-parameter group with
infinitesimal generator v = ~ x + rot + ¢~u (where ~,
r, ~odepend just on x, t and u). Solving for the parameter d, we would have
d=h(x,t,u,c)
=- - t +{[(x + c 2) + 3"y]/(u + 1/2/3)) 1/2 ,
ut +~u = 0 ,
u = e -xt cos(x/~x),
ott= Xo2,
10 February 1986
13, 3' constant,
which arises in the vibration of rods. If we set
u(x, t) = o ( t ) w ( x ) - 1]2/3,
arising from the side condition
UUxt - UxU t + Uxt/20 = O,
then we find that v and w satisfy the pair of ordinary
differential equations involving a separation constant
where h would necessarily be invariant under the
group, and hence satisfy
I~(h) = ~h x + rh t + tphu = 0
for all x, t, u, c. Since ~, r, ~odo not depend on the
parameter c, this would only be possible if, for each
fLxed x, t, u, the derivatives h x , ht, h u were linearly
dependent functions of c. A tedious computation
using wronskians shows that this is n o t the case, hence
the given two-parameter family of solutions cannot
come from a single local group of transformations.
(The same argument proves that any one-parameter
family of solutions does come from a local one-parameter group via the non-classical method of (8).
Moreover, if 3' = 0, so we are reduced to a nonlinear
wave equation, any one-parameter family of rational
solutions can be determined using the classical symmetry groups, of which there are several more in this
case.) A similar non.classical separation of variables
is discussed in ref. [10] for the equation of a nonlinear string.
111
Volume 114A, number 3
PHYSICS LETTERS
For general differential equations, one can now
easily envisage more general second order, and even
higher order side conditions to append in the hopes
of determining more general classes of solutions. Two
questions are apparent: (1) Which side conditions are
admissible in the sense that there do exist solutions
to the combined system of equations plus side con.
ditions? (2) Which side conditions are soluble in the
sense that the combined system is in some way easier
to solve than the original partial differential equation? We will not attempt to answer these questions
here, but merely provide two final examples to illustrate the possibilities which lie beyond simple groupinvariance and separability. First consider the twodimensional heat equation
(13)
u t = uxx + Uyy.
10 February 1986
Secondly, for the nonlinear wave equation
utt = ½ (U2)xx ,
equivalent to the system (6), we can generate a more
interesting class of solutions n o t obtainable by partial
invariance by appending the second order side condition
utt = 20t,
where t~ is a constant. The resulting over-determined
system is easy to solve, leading to the new explicit
solutions
u=at2+-x/~x+at+b,
= +-(t + a ) w / x + b,
a>0,
c~=O,
where a and b are arbitrary constants.
We append the third order side condition
Uxyt = 0 ,
(14)
(implying invariance under the group generated by
v = UxytOu), the general solution of which has the
"semi-separable" form
(15)
u ( x , y , t) = f ( x , y ) + g(x, t) + h(y, t ) .
Differentiating (13) and using (14), we see that u
must satisfy three partial differential equations
0 = Uxxxy + Uxyyy ,
Uxtt = Uxxxt,
Uytt = Uyyy t ,
each of which has one fewer independent variable
than the original eq. (13). A little manipulation shows
that the functions in (15) satisfy the equations
f x x + fyy = t~(x) - fl(y),
h t - hyy = ~ )
gt - gxx = T(t) - or(x),
- ~(t),
involving separation f u n c t i o n s (as opposed to separation constants) a,/3, and 7- One can now write down
a host of such solutions, much more general than the
analog of this "semi-separation" proceeds similarly
(and, indeed, is perhaps of even greater interest !).
112
References
[ 1 ] L.V. Ovsiannikov, Group analysis of differential equations (Academic Press, New York, 1982).
[2] S. Lie, Leipz. Berich. 1 (1895) 53; Gesammelte Abhandlungen, Vol. 4 (Teubner, Leipzig, 1929) pp. 320-384.
[3] G.W. Bluman and J.D. Cole, J. Math. Mech. 18 (1969)
1025.
[4] P.J. Olver and P. Rosenau, On the "non-classical method"
for gxoup-invariant solutions of differential equations,
preprint.
[5] E.G. Kainins and W. Miller, in: Prec. IUTAM-ISIMM
Symp. on Modern developments in analytic mechanics
(Turin, 1982) pp. 511-533.
[6] E.G. Kalnins and W. Miller, J. Math. Phys. 26 (1985)
1560.
[7] G.W. Bluman and J.D. Cole, Similarity methods for
differential equations, Appl. Math. Sci. 13 (Springer,
Berlin, 1974).
[8] R.L. Anderson and N.H. Ibragimov, Lie-B~/cklund
transformations in applications, SIAM Stud. Appl. Math.